Lin s Concordance Correlation Coefficient

Chapter 301

Lin’s Concordance Correlation

Coefficient

Introduction

This procedure calculates Lin’s concordance correlation coefficient (𝜌𝜌𝐶𝐶) from a set of bivariate data.

The statistic, ρ^c , is an index of how well a new test or measurement (Y) reproduces a gold standard test or measurement (X). It quantifies the agreement between these two measures of the same variable (e.g. chemical concentration). Like a correlation, ρ^c ranges from -1 to 1, with perfect agreement at 1. It cannot exceed the

absolute value of ρ, Pearson’s correlation coefficient between Y and X. It can be legitimately calculated on as few as ten observations.

The formulas used are found in an appendix of McBride (2005). The development of this statistic is presented Lin (1989, 1992, 2000), Lin, Hedayat, Sinha, and Yang (2002), and Lin, Hedayat, and Wu (2012).

Technical Details

Following Lin et al. (2002), assume that n observations Yk and Xk are selected from a bivariate population with means 𝜇𝜇_𝑌𝑌 and 𝜇𝜇_𝑋𝑋, variances 𝜎𝜎_𝑌𝑌² and 𝜎𝜎_𝑋𝑋², and correlation ρ (the Pearson correlation coefficient). Here, Y represents a measure from a candidate test or method and X represents the corresponding measure of the gold standard test or method.

The degree of concordance (agreement) between the two measures can be characterized by the expected value of their squared difference

( )

[

Y X

] ⁽

Y X

⁾

Y X Y X

E

−

²

= µ − µ

²

+ σ

²

+ σ

²

− 2 ρσ σ

. This is the expected squared perpendicular deviation from a 45º line through the origin.

If every pair from the bivariate population is in exact agreement, the above expectation would be 0. In order to create an index of concordance scaled to between -1 and 1, Lin used:

( )

[ ]

( )

[ ]

( )

( )

( )

^{2 2 2}

2 2 2

2 2 2

2 2

2 1 2

0 1 |

X Y X

Y

X Y

X Y X

Y

X Y X

Y X

Y

C E Y X

X Y E

σ σ µ

µ

σ ρσ

σ σ µ

µ

σ ρσ σ

σ µ

µ ρ ρ

+ +

= −

+ +

−

− + +

− −

=

=

−

− −

=

The value of 𝜌𝜌_𝐶𝐶 is estimated from a sample by 𝜌𝜌�_𝐶𝐶 where the usual sample counterparts are substituted into the above formula to obtain

𝜌𝜌�_𝐶𝐶 = 2𝑆𝑆𝑌𝑌𝑋𝑋

(𝑌𝑌� − 𝑋𝑋�)²+ 𝑆𝑆_𝑌𝑌²+ 𝑆𝑆_𝑋𝑋² where

𝑋𝑋� =1 𝑛𝑛 � 𝑋𝑋^𝑘𝑘

𝑛𝑛

𝑘𝑘=1

𝑌𝑌� =1 𝑛𝑛 � 𝑌𝑌^𝑘𝑘

𝑛𝑛

𝑘𝑘=1

𝑆𝑆_𝑋𝑋²= 1

𝑛𝑛 �(𝑋𝑋_𝑘𝑘− 𝑋𝑋�)²

𝑛𝑛

𝑘𝑘=1

𝑆𝑆_𝑌𝑌²=1

𝑛𝑛 �(𝑌𝑌𝑘𝑘− 𝑌𝑌�)²

𝑛𝑛

𝑘𝑘=1

𝑆𝑆_{𝑋𝑋𝑌𝑌}=1

𝑛𝑛 �(𝑋𝑋𝑘𝑘− 𝑋𝑋�)(𝑌𝑌_𝑘𝑘− 𝑌𝑌�)

𝑛𝑛

𝑘𝑘=1

In order to achieve a better approximation with the normal distribution, Lin (1989) transforms _{𝜌𝜌�𝐶𝐶}using Fisher’s z transformation to obtain

( ) ^.

1 ˆ 1 ˆ ˆ ln

ˆ tanh

2 1 1

 

 





−

= +

=

⁻

C C

C

ρ

ρ ρ λ

This quantity has an asymptotically normal distribution with mean

( ) _



 





−

= +

=

⁻

C C

C

ρ

ρ ρ

λ 1

ln 1 tanh

¹ ₂¹

and variance

( ) ( )

( ^{1 1} ) ² ( ^{( 1 1} ) ^{) 2} ( ¹ ) ^.

2 , 1

,

₂

2 2

4 4 2 2

2 3

2 2

2 2 2



 





 



− −

− + −

−

−

= −

C C C

C C C

C

n n

ρ ρ

υ ρ ρ

ρ

υ ρ ρ ρ

ρ ρ υ ρ

ρ σ

where

X Y

X Y

σ σ

µ

υ = µ ⁻

This variance is estimated using

( )

( ) ( ⁽ ) ⁾ ( ) _ ^{ }





 



− −

− + −

−

−

= −

₂

2 2

4 4 2 2

2 3

2 2

2 2 2

ˆ

2 1 ˆ

ˆ 1 ˆ

1 ˆ 2 ˆ 1 ˆ

1 ˆ 2 1

C C C

C C

C C

r u r

u r

r S n

ρ ρ ρ

ρ ρ

ρ ρ

λ

where

X YS S

X u Y

−

=

Note that we follow the advice of McBride and do not multiple u by (n-1)/n as he originally suggested.

A lower 100(1-α)% confidence limit can be calculated using the normal distribution as follows.

𝜌𝜌�𝐶𝐶,𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙= tanh�𝜆𝜆̂ − 𝑧𝑧𝛼𝛼𝑆𝑆_𝜆𝜆�� Other confidence intervals can be obtained similarly.

Missing Values

Rows with missing values in either of the variables are ignored.

Procedure Options

This section describes the options available in this procedure.

Variables Tab

This panel specifies the variables used in the analysis.

Variables

Y: First Measurement Variable

This variable contains the values of Y, the candidate measurement.

X: Second Measurement Variable

This variable contains the values of X, the gold standard measurement. This is the variance to which Y is compared.

In some situations, there is no gold standard. You just want to study the agreement of two measurement variables.

In that case, this is the second measurement variable.

Frequency Variable

Specify an optional frequency (count) variable here. This variable contains integers that represent the number of observations (frequency) associated with each row of the dataset.

If this option is left blank, each dataset row has a frequency of one. This variable lets you modify that frequency.

This may be useful when your data were previously tabulated and you want to enter counts.

Alpha Levels Alpha for C.I.’s

This is the value of alpha for the asymptotic confidence intervals of ρc. A 100 (1 – alpha)% confidence interval is calculated. Usually, this number will range from 0.1 to 0.001. A common choice for alpha is 0.05, which results in a 95% confidence interval.

Reports Tab

The following options control which reports and plots are displayed.

Select Reports

Run Summary ... Linear Regression

These options specify which reports are displayed.

Report Options

Precision

Specify the precision of numbers in the report. Single precision will display seven-place accuracy, while the double precision will display thirteen-place accuracy. Note that all reports are formatted for single precision only.

Variable Names

Specify whether to use variable names or (the longer) variable labels in report headings.

Decimal Places

Specify the number of digits after the decimal point to display on the output of values of this type. Note that this option in no way influences the accuracy with which the calculations are done.

Enter All to display all digits available. The number of digits displayed by this option is controlled by whether the Precision option is Single or Double.

Select Plots

Y vs X

These options control whether the scatter plot is displayed, its size, and its format.

Click the large plot format button to change the plot settings.

Example 1 – Comparing Two Measurements

In this example, we will compare a new quick measurement to an expensive, gold standard measure.

You may follow along here by making the appropriate entries or load the completed template Example 1 by clicking on Open Example Template from the File menu of the Lin’s Concordance Correlation Coefficient window.

1 Open the Lins CCC dataset.

• From the File menu of the NCSS Data window, select Open Example Data.

• Click on the file Lins CCC.NCSS.

• Click Open.

2 Open the Lin’s Concordance Correlation Coefficient window.

• Using the Analysis menu or the Procedure Navigator, find and select the Lin’s Concordance Correlation Coefficient procedure.

• On the menus, select File, then New Template. This will fill the procedure with the default template.

3 Specify the variables.

• On the procedure window, select the Variables tab.

• Double-click in the Y: First Measurement Variable box. This will bring up the variable selection window.

• Select Quick from the list of variables and then click Ok.

• Double-click in the X: Second Measurement Variable box. This will bring up the variable selection window.

• Select GoldStd from the list of variables and then click Ok.

4 Run the procedure.

• From the Run menu, select Run Procedure. Alternatively, just click the green Run button.

Run Summary Section

Parameter Value Parameter Value

Dependent Variable Quick Rows Processed 15

Independent Variable GoldStd Rows Used in Estimation 15

Frequency Variable None Rows with X Missing 0

R-Squared 0.9911 Rows with Freq Missing 0

Correlation 0.9956 Sum of Frequencies 15

Coefficient of Variation 0.0486 Mean Square Error, MSE 4.870055

√MSE 2.20682

This report shows information about which variables and rows were processed.

Lin’s Concordance Correlation Coefficient Section

Parameter Value

Concordance Correlation Coefficient, ρc 0.9953 Lower, One-Sided 95% C.L. of ρc 0.9885 Upper, One-Sided 95% C.L. of ρc 0.9984 Lower, Two-Sided 95% C.L. of ρc 0.9863 Upper, Two-Sided 95% C.L. of ρc 0.9984

Concordance Correlation Coefficient, ρc

The estimated value of Lin’s concordance correlation coefficient.

Lower, One-Sided 95% C.L. of ρc

This is the limit of a one-sided, 95% lower confidence interval for ρc. The interval is all values higher than this value. Thus, the confidence interval estimates that the true value is at least this value.

Upper, One-Sided 95% C.L. of ρc

This is the limit of a one-sided, 95% upper confidence interval for ρc. The interval is all values lower than this value. Thus, the confidence interval estimates that the true value is less than this value.

Two-Sided 95% C.L. of ρc

These are the limits of a two-sided, 95% confidence interval for ρc. The interval is all values between the upper and lower values. Thus, the confidence interval estimates that the true value is between these values.

Descriptive Statistics Section

Parameter Measurement 1 Measurement 2

Variable Quick GoldStd

Count 15 15

Mean 45.4 45

Standard Deviation 22.60468 22.36068

Minimum 12 10

Maximum 85 80

This report presents the usual descriptive statistics.

Linear Regression Section

Intercept Slope

Parameter B(0) B(1)

Regression Coefficients 0.1107 1.0064

Lower 95% C.L. -2.7337 0.9494

Upper 95% C.L. 2.9551 1.0634

Standard Error 1.3166 0.0264

T Value 0.0841 38.1562

Prob Level (T Test) 0.9343 0.0000

This report displays a brief summary of a linear regression of Y on X.

Scatter Plot

Scatter Plot

This scatter plot helps you to assess the agreement of the two measurements. The plot is shaded below the 45° line to make the assessment easier.